The Universe Is Expanding — Question 2

Question 2

Converting the Hubble rate exposes its characteristic time

For this exercise take $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, 1 Mpc=$3.086\times10^{19}$ km and 1 year=$3.156\times10^7$ s.

1. Convert H0 to inverse seconds and compute the Hubble time.

2. Find the recession rate for proper distance 50 Mpc in the nearby Hubble-flow approximation.

Question 2 - Solution

Converting the Hubble rate exposes its characteristic time

(a) Method and calculation

$$\boxed{H_0=70/(3.086\times10^{19})=2.268\times10^{-18}\ {\mathrm{s}}^{-1}}.$$$$\boxed{H_0^{-1}=4.409\times10^{17}\ {\mathrm{s}}=13.969\ {\mathrm{Gyr}}}.$$

(b) Interpretation and further calculation

$$v=H_0D.$$$$\boxed{v=70(50)=3500\ {\mathrm{km/s}}}.$$

Physical check

The Hubble time is a scale, not automatically the age. The present expansion rate alone does not specify expansion history.